1. The RHIC HBT Puzzle, Chiral Symmetry Restoration, and Pion Opacity John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA John G. Cramer (with Gerald A. Miller) University of Washington Seattle, Washington, USA ISMD 2005 Kromeriz, Czech Republic August 11, 2005

2. ISMD 2005, Kromeriz2 Primer 1: HBT Interferometry 1.Two identical pions will have a Bose-Einstein enhancement when their relative momentum ( q ) is small. 2.The 3-D momentum width of the BE enhancement  the 3-D size ( R ) of the pion “fireball” source. 3.Assuming complete incoherence, the “height” of the BE bump tells us the fraction ( ½ ) of pions participating in the BE enhancement. 4.The “out” radius of the source requires a pion energy difference  E  related to  (emission duration). C(q)-1 q (MeV/c) 1/R BE enhancement C(p 1,p 2 ) =  (p 1,p 2 )/  (p 1 )  (p 2 )

3. August 11, 2005ISMD 2005, Kromeriz3 beam direction HBT Momentum Geometry Relative momentum between pions is a vector  can extract 3D shape information p2p2 p1p1 q R long R side R out R long – along beam direction R out – along “line of sight”, includes time/energy information. R side –  “line of sight”, no time/energy information. Pre-RHIC expectations: (1) Large R o,s,l (~12-20 fm) (2) R out > R side by ~4 fm.

4. August 11, 2005ISMD 2005, Kromeriz4 The Featureless HBT Landscape The source radii, as inferred from HBT interferometry, are very similar over almost two orders of magnitude in collision energy. The ratio of R o /R s is near 1 at all energies, which naively implies a “hard” equation of state and explosive emission behavior (  ~0). AGS CERN RHIC The RHIC HBT Puzzle

5. August 11, 2005ISMD 2005, Kromeriz5 Primer 2: The Nuclear Optical Model 1.Divide the pions into “channels” and focus on pions (Channel 1) that participate in the BE correlation (about 60% of the spectrum pions). Omit “halo” and “resonance” pions and those converted to other particles (Channels 2, 3, etc.). 2.Solve the time-independent Klein-Gordon equation for the wave functions of Channel 1 pions, using a complex potential U. Im( U ) accounts for those pions removed from Channel 1. 3.The complex optical potential U does several things: (a) absorbs pions (opacity); (b) deflects pion trajectories (refraction, demagnification); (c) steals kinetic energy from the emerging pions; (d) produces Ramsauer-type resonances in the well, which can modulate apparent source size and emission intensity.

6. August 11, 2005ISMD 2005, Kromeriz6 Optical Wave Functions [|  | 2  (b)] Full Calculation K T = 197 MeV/c K T = 592 MeV/c K T = 25 MeV/c Imaginary Only Eikonal Approx. Observer

7. August 11, 2005ISMD 2005, Kromeriz7 The DWEF Formalism We use the Wigner distribution of the pion source current density matrix S 0 (x,K) (“the emission function”). The pions interact with the dense medium, producing S(x,K), the distorted wave emission function (DWEF): The  s are distorted (not plane) wave solutions of:, where U is the optical potential. Gyulassy et al., ‘79 Note: assumes chaotic pion sources. Correlation function: Distorted Waves

8. August 11, 2005ISMD 2005, Kromeriz8 The “Hydro-Inspired” Emission Function (Bose-Einstein thermal function) (medium density)

9. August 11, 2005ISMD 2005, Kromeriz9 Primer 3: Chiral Symmetry Question 1: The up and down “current” quarks have masses of 5 to 10 MeV. The   (a down + anti-up combination) has a mass of ~140 MeV. Where does the observed mass come from? Answer 1: The quarks are more massive in vacuum due to “dressing”. Also the pair is tightly bound by the color force into a particle so small that quantum-uncertainty zitterbewegung gives both quarks large average momenta. Part of the   mass comes from the kinetic energy of the constituent quarks. Question 2: What happens when a pion is placed in a hot, dense medium? Answer 2: Two things happen: 1.The binding is reduced and the pion system expands because of external color forces, reducing the zitterbewegung and the pion mass. 2.The quarks that were “dressed” in vacuum become “undressed” in medium, causing up, down, and strange quarks to become more similar and closer to massless particles, an effect called “chiral symmetry restoration”. In many theoretical scenarios, chiral symmetry restoration and the quark-gluon plasma phase usually go together. Question 3: How can a pion regain its mass when it goes from medium to vacuum? Answer 3: It must do work against an average attractive force, losing kinetic energy while gaining mass. In effect, it must climb out of a potential well that may be 140 MeV deep. medium vacuum

10. August 11, 2005ISMD 2005, Kromeriz10 The Chiral Symmetry Potential; [Son & Stephanov (2002)] Both terms of U are negative (attractive) U (b) =  (w 0 +w 2 p 2 )  (b), w 0 is real, w 2 is complex. screening mass  “velocity” Both v 2 and v 2 m 2  (T)  0 near T=T c. Dispersion relation for pions in nuclear matter.

11. August 11, 2005ISMD 2005, Kromeriz11 Parameters of the Model Thermal:T 0 (MeV),   (MeV) Space:R WS (fm), a WS (fm) Time:   (MeV/c),  (MeV/c) Flow:  f (#)  Optical Pot.:Re(w 0 ) (fm -2 ), Re(w 2 ) (#), Im(w 2 ) (#) Wave Eqn.:  (Kisslinger turned off for now) Total number of parameters: 11 (+1) Varied for the Cu+Cu prediction.

12. August 11, 2005ISMD 2005, Kromeriz12 DWEF Fits to STAR Data We have calculated pion wave functions in a partial wave expansion, applied them to a “hydro-inspired” pion source function, and calculated the HBT radii and spectrum. The correlation function C is calculated at q=30 MeV/c, about half way down the BE bump. (We do not use the 2 nd moment of C, which is unreliable.) We have fitted STAR data at  s NN =200 GeV, simultaneously fitting R o, R s, R l, and dN p /dy (fitting both magnitude and shape) at 8 momentum values (i.e., 32 data points), using a Levenberg- Marquardt fitting algorithm. In the resulting fit, the  2 per data point is ~2.2 and the  2 per degree of freedom is ~3.3. Only statistical (not systematic) errors are used in calculating  2. We remove long-lived “halo” resonance contributions to the spectrum (which are not included in the model) by multiplying the uncorrected spectrum by ½ (the HBT parameter) before fitting, then “un-correcting” the predicted spectrum with  ½.

13. August 11, 2005ISMD 2005, Kromeriz13 DWEF Fits to STAR 200 GeV Pion HBT Radii U=0 Re[U]=0 No flow Boltzmann Full Calculation Non-solid curves show the effects of turning off various parts of the calculation.

14. August 11, 2005ISMD 2005, Kromeriz14 DWEF Fit to STAR 200 GeV Pion Spectrum U=0 Re[U]=0 No flow Boltzmann Full Calculation Raw Fit Non-solid curves show the effects of turning off various parts of the calculation

15. August 11, 2005ISMD 2005, Kromeriz15 Meaning of the Parameters Temperature: 222 MeV  Chiral PT predicted at ~ 193 MeV Transverse flow rapidity: 1.6  v max = 0.93 c, v av = 0.66 c Mean expansion time: 8.1 fm/c  system expansion at ~ 0.5 c Pion emission between 5.5 fm/c and 10.8 fm/c  soft EOS. WS radius: 12.0 fm = R(Au) + 4.6 fm > R @ SPS WS diffuseness: 0.72 fm (similar to Low Energy NP experience) Re(U): 0.113 + 0.725 p 2  deep well  strong attraction. Im(U): 0.128 p 2  mfp  8 fm @ K T =1 fm -1  strong absorption  high density Pion chemical potential: m  =124 MeV, slightly less than mass(  ) We have evidence suggesting a CHIRAL PHASE TRANSITION!

16. August 11, 2005ISMD 2005, Kromeriz16 Potential-Off Radius Fits No Chemical or Optical Pot. No Optical No Real STAR Blast Wave Full Calculation Non-solid curves show the effects of refitting. Out Side Long R O /R S Ratio

17. August 11, 2005ISMD 2005, Kromeriz17 Potential-Off Spectrum Fits No Chemical or Optical Pot. No Real No Optical STAR Blast Wave Full Calculation Raw Fit Non-solid curves show the effects of potential- off refits. ModelChi^2Chi^2/#dataChi^2/#dof Full Calculation69.192.163.29 No Real Potential905.0928.2843.10 No Optical Potential1003.4431.3647.78 No Opt/ Chem Potential1416.6644.2767.46

18. August 11, 2005ISMD 2005, Kromeriz18 Low p T Ramsauer Resonances R O (fm) Pion Spectrum K T (MeV/c) R S (fm) Phobos 0-6% (preliminary) Raw Fit U=0 Re[U]=0 Boltzmann No flow Full Calculation |  (q, b)| 2  (b) at K T = 49.3 MeV/c For fit, would need =0.41

19. August 11, 2005ISMD 2005, Kromeriz19 200 GeV Cu+Cu Predictions Scale R WS, A WS, ,  by A 1/3 Scale R WS,  by A 1/3 Scale R WS, A WS, ,  by R elect Scale R WS,  by R elect STAR (preliminary) Conclusion: Space-time parameters (4) scale as A 1/3.

20. August 11, 2005ISMD 2005, Kromeriz20 Summary Quantum mechanics has solved the technical problems of applying opacity to HBT. We obtain excellent DWEF fits to STAR  s NN =200 GeV data, simultaneously fitting three HBT radii and the p T spectrum. The key is the deep real optical potential. The fit parameters are reasonable and indicate strong collective flow, significant opacity, and huge attraction. They describe pion emission in hot, highly dense matter with a soft pion equation of state. We have replaced the RHIC HBT Puzzle with evidence suggesting a chiral phase transition in RHIC collisions. We note that in most quark-matter scenarios, the QGP phase transition is usually accompanied by a chiral phase transition at about the same critical temperature.

21. August 11, 2005ISMD 2005, Kromeriz21 Outlook We have a new tool for investigating the presence (or absence) of chiral phase transitions in heavy ion collision systems. Its use requires both high quality pion spectra and high quality HBT analysis over a region that extends to fairly low momenta (K T ~150 MeV/c). We are presently attempting to “track” the CPT phenomenon to lower collision energies, where the deep real potential should presumably go away. We plan to try to replace the empirical emission function with a relativistic hydrodynamic calculation of the multidimensional phase space density. (DWEF  DWRHD)

22. The End A short paper (with erratum) describing this work has been published in Phys. Rev. Lett. 94, 102302 (2005); See ArXiv: nucl-th/0411031 ; A longer paper has been submitted to Phys. Rev. C; See ArXiv: nucl-th/0507004

23. Backup Slides

24. August 11, 2005ISMD 2005, Kromeriz24 Time-Independence, Resonances, and Freeze-Out We note that our use of a time-independent optical potential does not invoke the mean field approximation and is formally correct according to quantum scattering theory. (The semi-classical mind-set can be misleading.) While the optical potential is not time-dependent, some time- dependent effects can be manifested in the energy-dependence of the potential. (Time and energy are conjugate quantum variables.) An optical potential can implicitly include the effects of resonances, including heavy ones. Therefore, our present treatment implicitly includes resonances produced within the hot, dense medium. We note that more detailed quantum coupled-channels calculations could be done, in which selected resonances were treated as explicit channels coupled through interactions. Describing the present STAR data apparently does not require this kind of elaboration.

25. August 11, 2005ISMD 2005, Kromeriz25 Meaning of the Parameters Temperature: 222 MeV  Chiral PT predicted at ~ 193 MeV Transverse flow rapidity: 1.6  v max = 0.93 c, v av = 0.66 c Mean expansion time: 8.1 fm/c  system expansion at ~ 0.5 c Pion emission between 5.5 fm/c and 10.8 fm/c  soft EOS. WS radius: 12.0 fm = R(Au) + 4.6 fm > R @ SPS WS diffuseness: 0.72 fm (similar to Low Energy NP experience) Re(U): 0.113 + 0.725 p 2  deep well  strong attraction. Im(U): 0.128 p 2  mfp  8 fm @ K T =1 fm -1  strong absorption  high density Pion chemical potential: m  =124 MeV, slightly less than mass(  ) We have evidence suggesting a CHIRAL PHASE TRANSITION!

26. August 11, 2005ISMD 2005, Kromeriz26 Wave Equation Solutions We assume an infinitely long Bjorken tube and azimuthal symmetry, so that the (incoming) waves factorize: 3D  2D(distorted)  1D(plane) We solve the reduced Klein-Gordon wave equation: Partial wave expansion ! ordinary diff eq

27. August 11, 2005ISMD 2005, Kromeriz27 The Meaning of U Im (U) : Opacity, Re (U) :Refraction Pions lose energy and flux. Re(U) must exist: very strong attraction chiral phase transition Im[U 0 ]=- p   0,  1 mb,    = 1 fm -3, Im [U 0 ] = .15 fm -2,  = 7 fm

28. August 11, 2005ISMD 2005, Kromeriz28 Compute Correlation Function Correlation function is not Gaussian; we evaluate it near the q of experiment. The R 2 values are not the moments of the emission function S.

29. August 11, 2005ISMD 2005, Kromeriz29 Overview of the DWEF Model The medium is dense and strongly interacting, so the pions must “fight” their way out to the vacuum. This modifies their wave functions, producing the distorted waves used in the model. We explicitly treat the absorption of pions by inelastic processes (e.g., quark exchange and rearrangement) as they pass through the medium, as implemented with the imaginary part of an optical potential. We explicitly treat the mass-change of pions (e.g., due to chiral- symmetry breaking) as they pass from the hot, dense collision medium [m(  )  0]) to the outside vacuum [m(  )  140 MeV]. This is accomplished by solving the Klein-Gordon equation with an optical potential, the real part of which is a deep, attractive, momentum- dependent “mass-type” potential. We use relativistic quantum mechanics in a cylindrical geometry partial wave expansion to treat the behavior of pions producing Bose-Einstein correlations. We note that most RHIC theories are semi-classical, even though most HBT analyses use pions in the momentum region (p  < 600 MeV/c) where quantum wave-mechanical effects should be important. The model calculates only the spectrum of pions participating in the BE correlation (not those contributions to the spectrum from long-lived “halo” resonances, etc.).

30. August 11, 2005ISMD 2005, Kromeriz30 Semi-Classical Eikonal Opacity b l R Heiselberg and Vischer X +

31. August 11, 2005ISMD 2005, Kromeriz31 Influence of the Real Potential in the Eikonal Approximation Therefore the real part of U, no matter how large, has no influence here.

32. August 11, 2005ISMD 2005, Kromeriz32 Source De-magnification by the Real Potential Well Because of the mass loss in the potential well, the pions move faster there (red) than in vacuum (blue). This de-magnifies the image of the source, so that it will appear to be smaller in HBT measurements. This effect is largest at low momentum. n=1.00 n=1.33 A Fly in a Bubble Rays bend closer to radii V csr = (120 MeV) 2 Velocity in well Velocity in vacuum

33. August 11, 2005ISMD 2005, Kromeriz33 Correlation Functions (linear) OutSideLong K T = 100 MeV/c K T = 200 MeV/c K T = 400 MeV/c K T = 600 MeV/c

34. August 11, 2005ISMD 2005, Kromeriz34 Correlation Functions (log) OutSideLong K T = 100 MeV/c K T = 200 MeV/c K T = 400 MeV/c K T = 600 MeV/c

35. August 11, 2005ISMD 2005, Kromeriz35 |  ( , b)|   b ) at K T = 1.000 fm -1 = 197 MeV/c Wave Function of Full Calculation Imaginary Only Eikonal Observer